Stripe Soliton Research Articles

Dynamic behaviors of interaction solutions of (3+1)-dimensional Shallow Water wave equation

Rational solutions and interaction solutions of the (3+1)-dimensional Shallow Water wave equation are obtained depending on N-soliton solution by the Hirota bilinear method and long wave limit method. And breather, lump solutions and line rogue waves are obtained from two-soliton solution as well as their interaction solutions are derived from four-soliton solution. After the collision, breathers and lumps remain unchanged, but the line rogue wave has a difference. Furthermore, three-soliton solution is discussed, the interactions between some hybrid solutions, such as hybrid solution between the stripe soliton and breather, hybrid solution between stripe soliton and lump solution and so on, are demonstrated in detail. More interestingly, stripe soliton remain unchanged, but the amplitudes of the periodic line waves, lump solutions and rogue waves have a great change after the collision.

Lump and line soliton pairs to a [formula omitted]-dimensional integrable Kadomtsev–Petviashvili equation

By introducing the positive quadratic function, one kind of rational lump solution to a (2+1)-dimensional integrable Kadomtsev–Petviashvili (KP) equation is presented. By combining the positive quadratic function and an exponential function, we find the fusion and fission phenomena for two kinds of typical local excitations, which occur between a lump and a stripe soliton. Furthermore, by adding a hyperbolic function to the quadratic function, we have derived the solution with the interaction between a lump and line soliton pairs. The dynamical behaviors of such local solutions are discussed by choosing the appropriate parameters.

Adiabatic invariant analysis of dark and dark-bright soliton stripes in two-dimensional Bose-Einstein condensates

In the present work, we develop an adiabatic invariant approach for the evolution of quasi-one-dimensional (stripe) solitons embedded in a two-dimensional Bose-Einstein condensate. The results of the theory are obtained both for the one-component case of dark soliton stripes, as well as for the considerably more involved case of the two-component dark-bright (alias "filled dark") soliton stripes. In both cases, analytical predictions regarding the stability and dynamics of these structures are obtained. One of our main findings is the determination of the instability modes of the waves as a function of the parameters of the system (such as the trap strength and the chemical potential). Our analytical predictions are favorably compared with results of direct numerical simulations.

M-lump solutions to a [formula omitted]-dimensional nonlinear evolution equation

This paper aims at computing the M-lump solutions which decay to a uniform state in all directions for a (3+1)-dimensional nonlinear evolution equation. These solutions are constructed by taking a “long wave” limit of the corresponding N-soliton solutions obtained by direct methods. The dynamic properties of M-lump solutions describing multiple collisions of lumps are presented. In addition, we investigate the interaction between stripe solitons and lumps which is further discussed implying that lumps will be drowned or swallowed by the stripe solitons. Finally the dynamic properties of interactive wave solutions are graphically depicted by choosing the values of parameters.

Lump and interaction solutions to the [formula omitted]-dimensional Burgers equation

Based on the Hirota bilinear method, the lump solution of a (2+1)-dimensional Burgers equations is presented through symbolic computation with Maple, which is rationally localized in all directions in the space. Then the interaction solution between lump solution and one stripe solution is obtained and the result shows that the lump soliton will be drowned or swallowed by the stripe soliton, which can also be called the fission or fusion phenomenon. Furthermore, by the interaction between lump solution and a pair of resonance stripe solitons, a rogue wave phenomenon is revealed.

Characteristics of the solitary waves and lump waves with interaction phenomena in a (2 + 1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation

In this paper, we consider a ( $$2+1$$ )-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada (gCDGKS) equation, which is a higher-order generalization of the celebrated Kadomtsev–Petviashvili (KP) equation. By considering the Hirota bilinear form of the CDGKS equation, we study a type of exact interaction waves by the way of vector notations. The interaction solutions, which possess extensive applications in the nonlinear system, are composed by lump wave parts and soliton wave parts, respectively. Under certain conditions, this kind of solutions can be transformed into the pure lump waves or the stripe solitons. Moreover, we provide the graphical analysis of such solutions in order to better understand their dynamical behavior.

Parameter limit method and its application in the (4+1)-dimensional Fokas equation

A new Hirota’s bilinear form of the (4+1)-dimensional Fokas equation is obtained by an appropriate wave transformation. Applying the parameter limit method introduced in this paper, we obtain two homoclinic solutions and a rogue wave solution of (4+1)-dimensional Fokas equation. Besides, we also discuss the interaction between rogue wave solution and different forms of stripe solitons (one stripe soliton and double stripe solitons), and investigate the annihilation and emergence of rogue waves. Finally, the parametric reasons of soliton fusion and fission are discussed, furthermore, the fusion and fission phenomena of soliton are simulated by three-dimensional plots.

Lump and rogue waves for the variable-coefficient Kadomtsev–Petviashvili equation in a fluid

Under investigation in this paper is the variable-coefficient Kadomtsev–Petviashvili equation, which describes the long waves with small amplitude and slow dependence on the transverse coordinate in a single-layer shallow fluid. Employing the bilinear form and symbolic computation, we obtain the lump, mixed lump-stripe soliton and mixed rogue wave-stripe soliton solutions. Discussions indicate that the variable coefficients are related to both the lump soliton’s velocity and amplitude. Mixed lump-stripe soliton solutions display two different properties, fusion and fission. Mixed rogue wave-stripe soliton solutions show that a rogue wave arises from one of the stripe solitons and disappears into the other. When the time approaches 0, rogue wave’s energy reaches the maximum. Interactions between a lump soliton and one-stripe soliton, and between a rogue wave and a pair of stripe solitons, are shown graphically.

Lump solutions with interaction phenomena in the (2+1)-dimensional Ito equation

In this paper, we consider the (2+1)-dimensional Ito equation, which was introduced by Ito. By considering the Hirota’s bilinear method, and using the positive quadratic function, we obtain some lump solutions of the Ito equation. In order to ensure rational localization and analyticity of these lump solutions, some sufficient and necessary conditions are provided on the parameters that appeared in the solutions. Furthermore, the interaction solutions between lump solutions and the stripe solitons are discussed by combining positive quadratic function with exponential function. Finally, the dynamic properties of these solutions are shown via the way of graphical analysis by selecting appropriate values of the parameters.

Lump, periodic lump and interaction lump stripe solutions to the (2+1)-dimensional B-type Kadomtsev–Petviashvili equation

In this paper, the Hirota’s bilinear form is employed to investigate the lump, periodic lump and interaction lump stripe solutions of the (2+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation. Many results are obtained by dynamic process of figures. We analyze the propagation direction and horizontal velocity of lump solutions to find some constraint conditions which include positiveness and localization. In the process of the travel of the periodic lump solutions, it appears that the energy distribution is not symmetrical. The interaction lump stripe solutions of non-elastic indicate that the lump solitons are dropped and swallowed by the stripe soliton.

Interaction solutions for a reduced extended $$\mathbf{(3}\varvec{+}{} \mathbf{1)}$$(3+1)-dimensional Jimbo–Miwa equation

In this paper, the exact solutions of a reduced extended $$(3+1)$$ -dimensional Jimbo–Miwa equation are investigated with the help of its bilinear representation and symbolic computation. Firstly, a kind of bright–dark lump wave solutions is directly obtained by taking the solution F in bilinear equation as a quadratic function. Furthermore, the interaction solutions between one lump wave and one stripe wave are also presented by taking F as a combination of quadratic function and exponential function. Finally, by taking F as a combination of quadratic function and hyperbolic cosine function, the rogue wave which aroused by the interaction between lump soliton and a pair of stripe solitons are obtained. The dynamic properties of the above three kinds of exact solutions are displayed vividly by figures.

Quasi-one-dimensional spin-orbit- and Rabi-coupled bright dipolar Bose-Einstein-condensate solitons

We study the formation of stable bright solitons in quasi-one-dimensional (quasi-1D) spin-orbit- (SO-) and Rabi-coupled two pseudospinor dipolar Bose-Einstein condensates (BECs) of 164 Dy atoms in the presence of repulsive contact interactions. As a result of the combined attraction-repulsion effect of both interactions and the addition of SO and Rabi couplings, two kinds of ground states in the form of self-trapped bright solitons can be formed, a plane-wave soliton (PWS) and a stripe soliton (SS). These quasi-1D solitons cannot exist in a condensate with purely repulsive contact interactions and SO and Rabi couplings (no dipole). Neglecting the repulsive contact interactions, our findings also show the possibility of creating PWSs and SSs. When the strengths of the two interactions are close to each other, the SS develops an oscillatory instability indicating a possibility of a breather solution, eventually leading to its destruction. We also obtain a phase diagram showing regions where the solution is a PWS or SS.

Spatio-temporal dynamics and interaction of lump solutions for the (4+1)-D Fokas equation

The (4+1)-D Fokas equation is a new and important physical model. Its Hirota's bilinear form with a perturbation parameter is obtained by an appropriate trans-formation. A class of lump solutions and three different forms of spatio-temporal structure are obtained. Meanwhile, the theoretical analysis for the change of spatio-temporal structure is discussed by using the extreme value theory of multivariate function. Finally, the interaction between a stripe soliton and lump solution is discussed, and a new wave phenomenon that the lump solution is swallowed and drowned by the stripe soliton is investigated.

Abundant Lump Solutions and Interaction Phenomena to the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony Equation

In this paper, we obtained a kind of lump solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation with the assistance of Mathematica. Some contour plots with different determinant values are sequentially made to show that the corresponding lump solutions tend to zero when x2+y2→∞. Particularly, lump solutions with specific values of the include parameters are plotted, as illustrative examples. Finally, a combination of stripe soliton and lump soliton is discussed to the KP-BBM equation, in which such a solution presents two different interesting phenomena: lump-kink and lump-soliton. Simultaneously, breather rational soliton solutions are displayed.

Emergence and Interaction of the Lump-Type Solution with the (3+1)-D Jimbo-Miwa Equation

Abstract A kinky breather-soliton solution and kinky periodic-soliton solution are obtained using Hirota’s bilinear method and homoclinic test approach for the (3+1)-dimensional Jimbo-Miwa equation. Based on these two exact solutions, some lump-type solutions are emerged by limit behaviour. Meanwhile, two kinds of new dynamical phenomena, kinky breather degeneracy and kinky periodic degeneracy, are discussed and presented. Finally, the interaction between a stripe soliton and a lump-type soliton is discussed by the standardisation of the lump-type solution; the fusion and fission phenomena of soliton solutions are investigated and simulated by three-dimensional plots.

Moving bright solitons in a pseudo-spin polarization Bose–Einstein condensate**Project supported by the National Natural Science Foundation of China (Grant Nos. 11304270, 61774001, and 11475144) and the Key Project of Scientific and Technological Research of Hebei Province, China (Grant No. ZD2015133).

We study the moving bright solitons in the weak attractive Bose–Einstein condensate with a spin–orbit interaction. By solving the coupled nonlinear Schrödinger equation with the variational method and the imaginary time evolution method, two kinds of solitons (plane wave soliton and stripe solitons) are found in different parameter regions. It is shown that the soliton speed dominates its structure. The detuning between the Raman beam and energy states of the atoms decides the spin polarization strength of the system. The soliton dynamics is also studied for various moving speed and we find that the shape of individual components can be kept when the speed of soliton is low.

Deformation rogue wave to the (2+1)-dimensional KdV equation

Deformation rogue wave as exact solution of the (2+1)-dimensional Korteweg–de Vries (KdV) equation is obtained via the bilinear method. It is localized in both time and space and is derived by the interaction between lump soliton and a pair of resonance stripe solitons. In contrast to the general method to get the rogue wave, we mainly combine the positive quadratic function and the hyperbolic cosine function, and then the lump soliton can be evolved rogue wave. Under the small perturbation of parameter, rich dynamic phenomena are depicted both theoretically and graphically so as to understand the property of (2+1)-dimensional KdV equation deeply. In general terms, these deformations mainly have three types: two rogue waves, one rogue wave or no rogue wave.

A Pair of Resonance Stripe Solitons and Lump Solutions to a Reduced (3+1)-Dimensional Nonlinear Evolution Equation**Supported by National Natural Science Foundation of China under Grant Nos. 11271211, 11275072, and 11435005, Ningbo Natural Science Foundation under Grant No. 2015A610159 and the Opening Project of Zhejiang Provincial Top Key Discipline of Physics Sciences in Ningbo University under Grant No. xkzw11502 and K.C.

With symbolic computation, some lump solutions are presented to a (3+1)-dimensional nonlinear evolution equation by searching the positive quadratic function from the Hirota bilinear form of equation. The quadratic function contains six free parameters, four of which satisfy two determinant conditions guaranteeing analyticity and rational localization of the solutions, while the others are free. Then, by combining positive quadratic function with exponential function, the interaction solutions between lump solutions and the stripe solitons are presented on the basis of some conditions. Furthermore, we extend this method to obtain more general solutions by combining of positive quadratic function and hyperbolic cosine function. Thus the interaction solutions between lump solutions and a pair of resonance stripe solitons are derived and asymptotic property of the interaction solutions are analyzed under some specific conditions. Finally, the dynamic properties of these solutions are shown in figures by choosing the values of the parameters.

Lump Solutions and Interaction Phenomenon for (2+1)-Dimensional Sawada–Kotera Equation**Supported by the Global Change Research Program of China under Grant No. 2015CB953904, National Natural Science Foundation of China under Grant Nos. 11675054 and 11435005, Outstanding Doctoral Dissertation Cultivation Plan of Action under Grant No. YB2016039, and Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things under Grant No. ZF1213

In this paper, a class of lump solutions to the (2+1)-dimensional Sawada–Kotera equation is studied by searching for positive quadratic function solutions to the associated bilinear equation. To guarantee rational localization and analyticity of the lumps, some sufficient and necessary conditions are presented on the parameters involved in the solutions. Then, a completely non-elastic interaction between a lump and a stripe of the (2+1)-dimensional Sawada–Kotera equation is obtained, which shows a lump solution is drowned or swallowed by a stripe soliton. Finally, 2-dimensional curves, 3-dimensional plots and density plots with particular choices of the involved parameters are presented to show the dynamic characteristics of the obtained lump and interaction solutions.

Some Interaction Solutions of a Reduced Generalised (3+1)-Dimensional Shallow Water Wave Equation for Lump Solutions and a Pair of Resonance Solitons

Abstract Through Hirota bilinear transformation and symbolic computation with Maple, a class of lump solutions, rationally localised in all directions in the space, to a reduced generalised (3+1)-dimensional shallow water wave (SWW) equation are prensented. The resulting lump solutions all contain six parameters, two of which are free due to the translation invariance of the SWW equation and the other four of which must satisfy a nonzero determinant condition guaranteeing analyticity and rational localisation of the solutions. Then we derived the interaction solutions for lump solutions and one stripe soliton and the result shows that the particular lump solutions with specific values of the involved parameters will be drowned or swallowed by the stripe soliton. Furthermore, we extend this method to a more general combination of positive quadratic function and hyperbolic functions. Especially, it is interesting that a rogue wave is found to be aroused by the interaction between lump solutions and a pair of resonance stripe solitons. By choosing the values of the parameters, the dynamic properties of lump solutions, interaction solutions for lump solutions and one stripe soliton and interaction solutions for lump solutions and a pair of resonance solitons, are shown by dynamic graphs.